2. Field1 means and variances: space, time, spacetime 1. The GRAND (space+time) mean of my x and y: 27.5744 degC and 239.9729 (W/m^2). 2. The GRAND (space+time) variance of my x and y: 2.9326 (degC)^2 and 551.1539 (W/m^2)^2. 3. The GRAND standard deviations are: 1.7125 degC, 23.4770 W/m^2 4. The SPATIAL variance of my TIME MEAN longitude section is: 1.8427 (degC)^2 and 333.9493 (W/m^2)^2 5. The TEMPORAL variance of my LONGITUDE MEAN time series is: 0.3616 (degC)^2 and 23.7800 (W/m2)^2
3. (the assignment part) 1. Confirm that the time mean of the anomalies as defined above is 0. Yes 2. Is the spatial mean of the anomalies (as defined above) 0? No § o Is it the same as the time series of the spatial mean of the raw data? Or is it a new thing? 3. My CLIMATOLOGICAL ANNUAL CYCLE have variance: o
var( climx12(:),1) 2.4852 (sst) var( climy12(:),1) 410.5632 (olr) 4. My INTERANNUAL ANOMALY ARRAYS have variance: o var( anomx(:),1) 0.4473 var( anomy(:),1) 140.5907 5. Fill out a variance decomposition table for field 1: feel free to add columns if you can define other parts.
SST
a) total variance of x
2.9326
b) purely spatial (variance of TIME mean at each lon)
1.8427
c) variance of (x minus its TIME mean at each lon)
1.0898
d) purely temporal (variance of LON mean at each time)
0.3616
e) variance of (x minus its LON mean at each time)
2.5710
f) remove both means (space-time variability)
0.7282
g) mean seasonal cycle
2.4852
h) deseasonalized anomalies
0.4473
i) variance of longitudinal mean of h
0.1025
j) h minus i
0.3448
6. Discuss your results:
· (a) = (g) + (h). Yes it checks out.
4. Further decomposition of anomx by scale (using rebinning). o What space and time scales (units: degrees and months) have the most variance in your anomx field?
o Create a contour plot of anomx -- can you see these "characteristic" scales by eye? Annotate your anomx plot with some ovals of about the right size (in powerpoint may be easiest), and put a few of these ovals in regions where you think you can see structures of about the right scales. I am looking for good eyeball judgement here.
5. Scatter plot, correlation and covariance, regression-explained variance
Based on your data fields (which you've seen pictures of), make subsets of your 2 variables x and y and make a scatter plot of these showing the strongest (positive or negative) correlation of one field with the other you can find. The subset might simply be all (x,t) values if your fields are very similar (olr, precip), or maybe the 240 time values at one longitude, or 144 longitudinal values in the time mean, or time series at different longitudes if some variability is offset in your two fields (like pressure and wind).
Do a scatter plot at longitude 225
Now consider the covariance and correlation of the two subset arrays entering your scatterplot.
What is the correlation coefficient corresponding to this scatter plot?
1.0000 -0.7588
-0.7588 1.0000
What are the standard deviations of your two data subsets? std(x)=1.2902 stdev(y)=15.9138
What fraction of the variance of y can be 'explained' by linear regression on x (y = mx + b)? How does this relate to rho? How much y variance is explained? (variance: with units of y squared) What is m? Hint: these are simple questions: use the math formula, not a computer code (Hsieh section 1.4.2, Eq. 1.33).
The fraction of the variance of sst can be 'explained' by linear regression on olr is rho.^2=0.5758.
The explained variance of sst is std(x)^2*rho.^2=0.9584
m is the slope of the least-squares fit line rho*std(y)/std(x)=9.359
What fraction of the variance of x can be 'explained' by linear regression on variable y? (x = nx + a)? How does this relate to rho? What is n? Hint: these are simple questions, use the math formula not computer code.
Now add uncorrelated (random) noise with variance 1 to one of your variables. This might be like observation error. (In this case, it was added to OLR)
How did the variance of y change when this noise was added?
var(y(:))=551.1699, var(noisey(:))=552.4670
Tried doing it for SST variable also, we get var(x(:))=2.9326, var(noisex(:))=3.9302
My original OLR variance @ 225° was 253.249. The variance of my sst + random noise is: 255.2928. The addition of noise increases the variance.
How did the correlation change? rho = corrcoef(x(:),noisey(:))
1.0000 -0.4975
-0.4975 1.0000
@ 255 degrees lon:
1.0000 -0.7607
-0.7607 1.0000
How do these changes affect the regression of y on x? How much (y+noise) variance is explained by linear regression on x? What is the new value of m in the new (noisey = mx + b) regression?
Hint: all these could be answered without using the computer, but it may help to confirm with data
std(noisey)=15.9138
rho^2 = 0.7607^2 = 0.5786
slope m, rho*std(y)/std(x)=9.382
6. Lagged correlation, covariance, and cross-covariance: hey let's compute all vs. all
Show the zero-lag spatial covariance and correlation structures for your primary field, like OLR_anoms_covar_correl.BEM.Matlab.png this for OLR. (please label the axes better than I did!) Interpret the results.
just did the OLR for fun
largest variance seen in SST is from 150-300 longitude (need to take x axis and multiply by 2.5 because it was scaled to 0-144).
For OLR, the largest variance seen is from 150-250 longitude.
For the zero-lag correlation plots, SST and OLR have similar graphs with a negative correlation from around 150-250 longitude which may indicate to us el nino/la nina events happening
Show longitude-lag sections of the covariance or correlation of this field, for a base point at some longitude of interest. Like this for OLR at a central Pacific longitude: OLR.lagregression.BEM.jpg (Please label the axes better than I did in this example! I hate Matlab). Better in IDL: olr_lag_covariances.gif
Intepret the results in terms of the characteristic space and time scales of your anomalies. Can you see these characteristic scales in your original raw data like in [[file/view/olr_lag_covariances.gif|olr_lag_covariances.gif]]?
Share a longitude-lag slice of your lagged co-variance matrix for your TWO fields. Label it, interpret it.
Elizabeth Wong
2. Field1 means and variances: space, time, spacetime
1. The GRAND (space+time) mean of my x and y: 27.5744 degC and 239.9729 (W/m^2).
2. The GRAND (space+time) variance of my x and y: 2.9326 (degC)^2 and 551.1539 (W/m^2)^2.
3. The GRAND standard deviations are: 1.7125 degC, 23.4770 W/m^2
4. The SPATIAL variance of my TIME MEAN longitude section is: 1.8427 (degC)^2 and 333.9493 (W/m^2)^2
5. The TEMPORAL variance of my LONGITUDE MEAN time series is: 0.3616 (degC)^2 and 23.7800 (W/m2)^2
3. (the assignment part)
1. Confirm that the time mean of the anomalies as defined above is 0. Yes
2. Is the spatial mean of the anomalies (as defined above) 0? No
§
o Is it the same as the time series of the spatial mean of the raw data? Or is it a new thing?
3. My CLIMATOLOGICAL ANNUAL CYCLE have variance:
o
var( climx12(:),1) 2.4852 (sst)
var( climy12(:),1) 410.5632 (olr)
4. My INTERANNUAL ANOMALY ARRAYS have variance:
o var( anomx(:),1) 0.4473
var( anomy(:),1) 140.5907
5. Fill out a variance decomposition table for field 1: feel free to add columns if you can define other parts.
SST
a) total variance of x
2.9326
b) purely spatial (variance of TIME mean at each lon)
1.8427
c) variance of (x minus its TIME mean at each lon)
1.0898
d) purely temporal (variance of LON mean at each time)
0.3616
e) variance of (x minus its LON mean at each time)
2.5710
f) remove both means (space-time variability)
0.7282
g) mean seasonal cycle
2.4852
h) deseasonalized anomalies
0.4473
i) variance of longitudinal mean of h
0.1025
j) h minus i
0.3448
· (a) = (g) + (h). Yes it checks out.
4. Further decomposition of anomx by scale (using rebinning).
o What space and time scales (units: degrees and months) have the most variance in your anomx field?
§ Matlab prints it:
§ variance_by_scalefactor =
0.4473 0.4428 0.4333 0.4167 0.3826 0.1663
§ 0.4221 0.4187 0.4112 0.3976 0.3683 0.1604
§ 0.3860 0.3835 0.3780 0.3672 0.3431 0.1497
§ 0.3265 0.3247 0.3208 0.3130 0.2949 0.1305
§ 0.1874 0.1864 0.1841 0.1796 0.1697 0.0755
§ 0.0504 0.0500 0.0493 0.0475 0.0447 0.0178
o Create a contour plot of anomx -- can you see these "characteristic" scales by eye? Annotate your anomx plot with some ovals of about the right size (in powerpoint may be easiest), and put a few of these ovals in regions where you think you can see structures of about the right scales. I am looking for good eyeball judgement here.
5. Scatter plot, correlation and covariance, regression-explained variance
6. Lagged correlation, covariance, and cross-covariance: hey let's compute all vs. all